This invention relates to a method and apparatus for attitude control of a spacecraft in orbit. It is known to provide attitude control for spacecraft in a geosynchronous equatorial plane orbit to correct for roll and yaw errors. For communication satellites, maintenance of a geosynchronous equatorial orbit greatly simplifies attitude control to obtain necessary pointing accuracy for the spacecraft.
Maintenance of this equatorial orbit necessarily expends considerable propellant as spacecraft in equatorial orbit are subject to various destabilizing forces. These destabilizing forces include gravitational effects from the sun and moon which alter orientation of the geosynchronous orbit from an equatorial orbit with zero inclination, to a slightly inclined orbit. Stationkeeping is a function implemented by the spacecraft to maintain the spacecraft at a particular inclination. As more stationkeeping maneuvers are required, more propellant is used by the spacecraft, resulting in a determinable cost per operational lifetime. As an amount of propellant is increased to provide for extended stationkeeping, this cost necessarily increases. There is a well-recognized trade-off between the amount of propellant needed to fulfil stationkeeping maneuvering over the lifetime of the spacecraft versus its operational life. As more propellant is needed, the more cost is associated with putting the spacecraft into orbit. Similarly, as operational lifetime of the spacecraft is increased, additional propellant is required which again adds to the spacecraft's cost. It is therefore a desire associated with existing spacecraft to extend operational life without significantly increasing cost.
One solution is to operate the spacecraft in its geosynchronous equatorial orbit until propellant is substantially exhausted. Thereafter, the spacecraft enters into an inclined orbit due to the disturbance forces. Generally drift of the inclination of the spacecraft's orbit is at a rate of about 0.8 degrees to about 0.9 degrees annually.
Unfortunately, as inclination of the spacecraft increases, maintenance of a particular desired pointing accuracy becomes increasingly more difficult. This difficulty in maintaining a desired pointing accuracy is further increased by the lack of propellant which necessitated operation of the spacecraft in the inclined orbit.
It is known that at a particular inclination, roll, pitch and yaw errors in spacecraft pointing are introduced. In an equatorial orbit, the spacecraft has an aim point which nominally points to the nadir, which is the equator. As inclination of the spacecraft increases, the aim point is no longer necessarily the equator. The spacecraft pointing traces out a figure eight with respect to a particular desired aim point due to the roll and pitch errors.
Generally, for a 5-degree inclination orbit, north-south excursion of the figure eight is approximately 10 degrees, and east-west excursion is approximately .+-.0.02 or 0.04 degrees. These excursions are unacceptable deviations for many missions of the spacecraft.
Method and apparatus are known for estimation of yaw and other disturbance torques to permit accurate control of yaw errors in an orbiting satellite. Systems implementing these methods often incorporate orbital dynamics specific to equatorial orbits of geosynchronous spacecraft such that operation of the systems cannot be used accurately in an inclined orbit. The prior art has known the use of momentum bias spacecraft, for example, an L-wheel system configured spacecraft or a V-wheel system. Also known are techniques which have been discussed for control of a geosynchronous spacecraft in an inclined orbit. Systems operable in an inclined orbit attempt to compensate for inclination-induced pointing inaccuracies by introducing a relatively slow periodic sinusoidal disturbance into applicable roll commands. The sinusoid disturbance typically has a period of one day, which in essence causes the pointing of the spacecraft to "nod" at the one-day frequency. Nodding of the spacecraft compensates for so-called gross inclination errors which are generally viewed as the north-south excursions. In many instances, compensation of only these gross errors is insufficient for precise pointing required in, for example, narrow beam communications.
An example of a V-wheel momentum bias spacecraft system was disclosed in U.S. Pat. No. 4,521,855 issued to Lehner et al. hereby expressly incorporated by reference for all purposes. In this V-wheel system, two momentum wheels were oriented with respect to one another and the spacecraft such that a total momentum was established in a pitch-yaw plane. Spacecraft roll and pitch were controlled in a known manner by selective control of each of the wheels of the V-wheel system. A wheel speed which is related to a momentum of each wheel, as well understood, was alternately increased in a first wheel while decreased in a second wheel to effect roll of the spacecraft. To effect pitch, speed of both wheels was increased or decreased simultaneously. Yaw error correction was passively implemented by a quarter-orbit coupling as is known in the art.
As a result of orbital dynamics, typical V-wheel systems manifest a phenomenon wherein the momentum wheels continuously gain greater speed due to external disturbance such that momentum unload is required to maintain control. It is known to use magneto-torquers or thrusters as actuators to implement momentum unloads oriented in a yaw/roll plane.
Equations of motion for momentum bias spacecraft have been derived for small inclined orbits. The derivation of these equations will be described next. Equations of motion of the spacecraft in an inclined orbit were initially formulated as transfer functions for a spacecraft having a single pitch wheel. Roll/yaw and pitch dynamics were decoupled from each other, allowing the dynamics to be addressed individually. Equations of motion for a system with both pitch and yaw momentum storage, particularly for roll/yaw, ignoring products of inertia, are as follows: ##EQU1## where: ##EQU2## where .phi. and .psi. are spacecraft roll and yaw angles; H.sub.x and H.sub.z are body axis roll and yaw angular momentum components; h.sub.z is a total yaw momentum stored in all of the wheels; h.sub.zc is a commanded wheel yaw momentum; I.sub.x, I.sub.y, and I.sub.z are principal moments of inertia about roll, pitch, and yaw axes, respectively; .omega..sub.0 &gt;0 and denominated as an orbital rate for the system H.sub.n &gt;0, and is denominated the momentum bias; .tau. is a time constant of momentum wheels; and M.sub.x and M.sub.z are body axis torques. Equation 1 is applicable for all wheel configurations in which there is no roll momentum storage.
In order to eliminate roll ground track errors, a spacecraft must track a desired roll trajectory. The yaw momentum storing wheels are controlled to minimize nutation and allow for roll tracking. Momentum wheels are unloaded by external torque actuators. With an assumption that nutation dynamics are damped by a wheel controller, Equation 1 is simplified to: ##EQU3## As nutation frequencies of most spacecraft are greater than typical orbit rates, Equation 5 accurately represents an orbit rate transient response and a low frequency disturbance response of a spacecraft attitude. The angular momentum of a spacecraft is related to attitude errors and wheel yaw momentum as follows: EQU H.sub.z =h.sub.z -H.sub.n .phi. (6)
and EQU H.sub.x =H.sub.n .psi. (7)
The spacecraft is erected with a momentum perpendicular to an equatorial plane instead of to an inclined orbital plane to passively provide a desired yaw angle for the spacecraft. To obtain a momentum perpendicular to the equatorial plane, a wheel spin-up maneuver is performed at an antinode. Under thruster control, momentum wheels are "spun-up" to different speeds to generate a desired yaw momentum H.sub.zi, which places a total system momentum (no yaw error) perpendicular to an equatorial orbit plane. A quarter of an orbit later, at a line of nodes H.sub.zi transforms to yaw angle .psi. as demonstrated by Equation 5. This transformation of H.sub.zi to yaw angle .psi. allows for passive generation of a desired yaw angle .psi..sub.i. Thrusters, or magnetic torquers, regulate a system momentum about this momentum configuration. Alternate methods of wheel spin-up are possible with yaw sensing.
A yaw momentum required to generate .psi..sub.i is as follows: EQU H.sub.zi =-i(t)H.sub.n sin (.omega..sub.o t+.kappa.(t)) (8)
resulting in EQU .psi..sub.i =i(t) cos (.omega..sub.0 t+.kappa.(t)) (9)
the desired yaw angle. Here, i(t) and .kappa.(t) are a time varying inclination and time varying position of an ascending node, respectively. An object of a control system of the present invention is to force an actual yaw momentum .psi..sub.i of a spacecraft to track H.sub.zi.
As indicated above, pitch dynamics are decoupled from roll/yaw dynamics and are expressed in the following form: ##EQU4## where .theta. is pitch angle and M.sub.y is an externally applied pitch axis torque, h.sub.y is an equivalent wheel pitch momentum, and h.sub.yc is a commanded pitch momentum.
There are three objectives of this control system:
1. To provide for yaw angle tracking by controlling a total yaw momentum of a spacecraft H.sub.zi. Total yaw momentum of the spacecraft is controlled through use of external torque actuators such as magnetic torquers or thrusters.
2. To provide for roll angle tracking by varying a distribution of yaw momentum between a roll angle .phi..sub.i and a yaw momentum in the wheels h.sub.z. Distribution of yaw momentum is controlled by varying a speed of a momentum wheel through the yaw momentum command h.sub.zc ; and
3. To provide for a pitch angle tracking .theta..sub.i by varying a pitch momentum of a spacecraft. Pitch momentum is controlled by varying a speed of a momentum wheel through a pitch momentum command h.sub.yc.
FIG. 1 illustrates an environment for a spacecraft 10 in an inclined orbit 25 about Earth 20 at an ascending node 22. An equatorial plane 21 and an orbit plane 23 are skewed along a line of nodes 28 by an inclination i. An orbiting reference frame x-y-z is established for the spacecraft 10 orbit normal 11 having a roll axis x pointing along a direction of motion, a pitch axis y perpendicular to the orbital plane 23, and a yaw axis z pointing towards the center of the Earth 20. Roll, pitch and yaw angles are measured relative to this reference frame x-y-z using standard roll-pitch-yaw Euler rotations.
The spacecraft 10 rotates about a negative pitch axis -y once per day with rotation rate .omega..sub.0. Momentum wheels (not shown) onboard the spacecraft 10 provide a momentum bias H.sub.n, also along the negative pitch axis -y.
Attitude errors induced by the inclination i of the orbit 25 arise when the spacecraft 10 is not pointing as it would in a nominal equatorial orbit. There are two major sources of inclination attitude errors: reference frame misalignment and ground tracking errors. When an orbit is inclined, the orbiting reference frame x-y-z is no longer aligned with the equator. This relative misalignment of the orbiting reference frame results in a time-varying yaw angle .psi..sub.i between the inclined orbit reference frame x-y-z and an equatorial orbit reference frame. The yaw angle .psi..sub.i angle is maximum at the ascending node 22 and the descending node 24 and is zero at antinodes 29 located between the ascending node 22 and the descending node 24. FIG. 1 illustrates attitude of the spacecraft 10 at four positions around the orbit 25.
Ground tracking errors arise because the spacecraft 10 is not positioned in the equatorial plane 23, and the yaw axis line of sight or nominal pointing 11 does not intersect the Earth 20 at the equator 14. FIG. 2 illustrates a yaw axis pointing direction of spacecraft 10 at an antinode 29. The spacecraft 10 is above the equatorial plane 23 having nominal pointing 11 of its yaw axis intersecting the Earth 20 at a point above the equator 14 (nadir 16). The spacecraft 10 must be biased downward in roll to point at the same point 26 on the equator 24 at which it would have pointed had the spacecraft 10 been in an equatorial orbit. To compensate for pitch errors, the spacecraft must be biased in pitch at other points in the orbit. The ground track 27, or intersection of the yaw axis and the Earth 20 surface, is a figure eight which is traced out with pitch errors and roll errors as the spacecraft 10 proceeds around the orbit 25.
The width of the ground track 27 figure eight is also related to the inclination i of the orbit 25, and the spacecraft 10 must compensate for resulting pitch errors. What is needed is a method and an apparatus for compensating for such resulting errors which minimizes fuel expenditure and maintains a particular level of autonomy.